We use COMSOL Multiphysics to solve time-dependent optimal control problems for partial differential equations whose optimality conditions can be formulated as a PDE. For a class of linear-quadratic model problems we summarize known analytic results on existence of solutions and first order optimality conditions that exhibit the typical feature of time-dependent control problems, namely the fact that a part of the optimality system has to be
integrated backward in time.
We present a strategy that is based on the treatment of the
coupled optimality system in the space-time cylinder. A brief motivation of this approach is given by showing that the optimality system is elliptic in some sence.
Numerical examples show advantages and limits of the usage of COMSOL Multiphysics and of our approach.